Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for realworld networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs.With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. Instead of studying directly hyperbolic random graphs, we use a generalization that we call geometric inhomogeneous random graphs (GIRGs). Since we ignore constant factors in the edge probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by this new model in future theoretical studies.We prove the following fundamental structural and algorithmic results on GIRGs.(1) As our main contribution we provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a substantial factor O( √ n).(2) We establish that GIRGs have clustering coefficients in Ω(1), (3) we prove that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits. * We choose a toroidal ground space for the technical simplicity that comes with its symmetry and in order to obtain hyperbolic random graphs as a special case. The results of this paper stay true if T d is replaced, say, by the d-dimensional unitcube [0, 1] d . † A major difference between hyperbolic random graphs and our generalisation is that we ignore constant factors in the edge probabilities puv. This allows to greatly simplify the edge probability expressions, thus reducing the technical overhead. W [20,21]. Note that the term min{1, .} is necessary, as the product w u w v may be larger than W. Classically, the Θ simply hides a factor 1, but by ‡ We say that an event holds with high probability (whp) if it holds with probability 1 − n −ω(1) .

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The algorithmic small-world phenomenon, empirically established by Milgram's letter forwarding experiments from the 60s [59], was theoretically explained by Kleinberg in 2000 [46]. However, from today's perspective his model has several severe shortcomings that limit the applicability to real-world networks. In order to give a more convincing explanation of the algorithmic small-world phenomenon, we study decentralized greedy routing in a more flexible random graph model (geometric inhomogeneous random graphs) which overcomes all previous shortcomings. Apart from exhibiting good properties in theory, it has also been extensively experimentally validated that this model reasonably captures real-world networks.In this model, the greedy routing protocol is purely distributed as each vertex only needs to know information about its direct neighbors. We prove that it succeeds with constant probability, and in case of success almost surely finds an almost shortest path of length Θ(log log n), where our bound is tight including the leading constant. Moreover, we study natural local patching methods which augment greedy routing by backtracking and which do not require any global knowledge. We show that such methods can ensure success probability 1 in an asymptotically tight number of steps.These results also address the question of Krioukov et al. [51] whether there are efficient local routing protocols for the internet graph. There were promising experimental studies, but the question remained unsolved theoretically. Our results give for the first time a rigorous and analytical affirmative answer. *

An edge-weighting of a graph is called vertex-coloring if the weighted degrees yield a proper vertex coloring of the graph. It is conjectured that for every graph without isolated edge, a vertex-coloring edge-weighting with the set {1, 2, 3} exists. In this note, we show that the statement is true for the weight set {1, 2, 3, 4}.

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Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if at the end, all the vertices are colored. The game chromatic number χg(G) is defined as the smallest k for which the first player has a winning strategy. Recently, Bohman, Frieze and Sudakov [Random Structures and Algorithms 2008] analysed the game chromatic number of random graphs and obtained lower and upper bounds of the same order of magnitude. In this paper we improve existing results and show that with high probability, the game chromatic number χg(Gn,p) of dense random graphs is asymptotically twice as large as the ordinary chromatic number χ(Gn,p).

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